Practical Bayesian Optimization of Machine

Learning Algorithms

Jasper Snoek

Department of Computer Science

University of Toronto

jasper@cs.toronto.edu

Hugo Larochelle

Department of Computer Science

University of Sherbrooke

hugo.larochelle@usherbrooke.edu

Ryan P.Adams

School of Engineering and Applied Sciences

Harvard University

rpa@seas.harvard.edu

Abstract

The use of machine learning algorithms frequently involves careful tuning of

learning parameters and model hyperparameters.Unfortunately,this tuning is of-

ten a “black art” requiring expert experience,rules of thumb,or sometimes brute-

force search.There is therefore great appeal for automatic approaches that can

optimize the performance of any given learning algorithmto the problemat hand.

In this work,we consider this problem through the framework of Bayesian opti-

mization,in which a learning algorithm’s generalization performance is modeled

as a sample from a Gaussian process (GP).We show that certain choices for the

nature of the GP,such as the type of kernel and the treatment of its hyperparame-

ters,can play a crucial role in obtaining a good optimizer that can achieve expert-

level performance.We describe newalgorithms that take into account the variable

cost (duration) of learning algorithm experiments and that can leverage the pres-

ence of multiple cores for parallel experimentation.We show that these proposed

algorithms improve on previous automatic procedures and can reach or surpass

human expert-level optimization for many algorithms including latent Dirichlet

allocation,structured SVMs and convolutional neural networks.

1 Introduction

Machine learning algorithms are rarely parameter-free:parameters controlling the rate of learning

or the capacity of the underlying model must often be speciﬁed.These parameters are often con-

sidered nuisances,making it appealing to develop machine learning algorithms with fewer of them.

Another,more ﬂexible take on this issue is to view the optimization of such parameters as a proce-

dure to be automated.Speciﬁcally,we could view such tuning as the optimization of an unknown

black-box function and invoke algorithms developed for such problems.A good choice is Bayesian

optimization [1],which has been shown to outperform other state of the art global optimization

algorithms on a number of challenging optimization benchmark functions [2].For continuous func-

tions,Bayesian optimization typically works by assuming the unknown function was sampled from

a Gaussian process and maintains a posterior distribution for this function as observations are made

or,in our case,as the results of running learning algorithm experiments with different hyperpa-

rameters are observed.To pick the hyperparameters of the next experiment,one can optimize the

expected improvement (EI) [1] over the current best result or the Gaussian process upper conﬁdence

bound (UCB)[3].EI and UCB have been shown to be efﬁcient in the number of function evaluations

required to ﬁnd the global optimumof many multimodal black-box functions [4,3].

1

Machine learning algorithms,however,have certain characteristics that distinguish themfromother

black-box optimization problems.First,each function evaluation can require a variable amount of

time:training a small neural network with 10 hidden units will take less time than a bigger net-

work with 1000 hidden units.Even without considering duration,the advent of cloud computing

makes it possible to quantify economically the cost of requiring large-memory machines for learn-

ing,changing the actual cost in dollars of an experiment with a different number of hidden units.

Second,machine learning experiments are often run in parallel,on multiple cores or machines.In

both situations,the standard sequential approach of GP optimization can be suboptimal.

In this work,we identify good practices for Bayesian optimization of machine learning algorithms.

We argue that a fully Bayesian treatment of the underlying GP kernel is preferred to the approach

based on optimization of the GP hyperparameters,as previously proposed [5].Our second contri-

bution is the description of newalgorithms for taking into account the variable and unknown cost of

experiments or the availability of multiple cores to run experiments in parallel.

Gaussian processes have proven to be useful surrogate models for computer experiments and good

practices have been established in this context for sensitivity analysis,calibration and prediction [6].

While these strategies are not considered in the context of optimization,they can be useful to re-

searchers in machine learning who wish to understand better the sensitivity of their models to various

hyperparameters.Hutter et al.[7] have developed sequential model-based optimization strategies for

the conﬁguration of satisﬁability and mixed integer programming solvers using randomforests.The

machine learning algorithms we consider,however,warrant a fully Bayesian treatment as their ex-

pensive nature necessitates minimizing the number of evaluations.Bayesian optimization strategies

have also been used to tune the parameters of Markov chain Monte Carlo algorithms [8].Recently,

Bergstra et al.[5] have explored various strategies for optimizing the hyperparameters of machine

learning algorithms.They demonstrated that grid search strategies are inferior to randomsearch [9],

and suggested the use of Gaussian process Bayesian optimization,optimizing the hyperparameters

of a squared-exponential covariance,and proposed the Tree Parzen Algorithm.

2 Bayesian Optimization with Gaussian Process Priors

As in other kinds of optimization,in Bayesian optimization we are interested in ﬁnding the mini-

mum of a function f(x) on some bounded set X,which we will take to be a subset of R

D

.What

makes Bayesian optimization different from other procedures is that it constructs a probabilistic

model for f(x) and then exploits this model to make decisions about where in X to next evaluate

the function,while integrating out uncertainty.The essential philosophy is to use all of the informa-

tion available from previous evaluations of f(x) and not simply rely on local gradient and Hessian

approximations.This results in a procedure that can ﬁnd the minimumof difﬁcult non-convex func-

tions with relatively few evaluations,at the cost of performing more computation to determine the

next point to try.When evaluations of f(x) are expensive to perform — as is the case when it

requires training a machine learning algorithm —then it is easy to justify some extra computation

to make better decisions.For an overview of the Bayesian optimization formalism and a review of

previous work,see,e.g.,Brochu et al.[10].In this section we brieﬂy review the general Bayesian

optimization approach,before discussing our novel contributions in Section 3.

There are two major choices that must be made when performing Bayesian optimization.First,one

must select a prior over functions that will express assumptions about the function being optimized.

For this we choose the Gaussian process prior,due to its ﬂexibility and tractability.Second,we

must choose an acquisition function,which is used to construct a utility function from the model

posterior,allowing us to determine the next point to evaluate.

2.1 Gaussian Processes

The Gaussian process (GP) is a convenient and powerful prior distribution on functions,which we

will take here to be of the formf:X!R.The GP is deﬁned by the property that any ﬁnite set of N

points fx

n

2 Xg

N

n=1

induces a multivariate Gaussian distribution on R

N

.The nth of these points

is taken to be the function value f(x

n

),and the elegant marginalization properties of the Gaussian

distribution allow us to compute marginals and conditionals in closed form.The support and prop-

erties of the resulting distribution on functions are determined by a mean function m:X!R and

a positive deﬁnite covariance function K:X X!R.We will discuss the impact of covariance

functions in Section 3.1.For an overviewof Gaussian processes,see Rasmussen and Williams [11].

2

2.2 Acquisition Functions for Bayesian Optimization

We assume that the function f(x) is drawn from a Gaussian process prior and that our observa-

tions are of the form fx

n

;y

n

g

N

n=1

,where y

n

N(f(x

n

);) and is the variance of noise intro-

duced into the function observations.This prior and these data induce a posterior over functions;

the acquisition function,which we denote by a:X!R

+

,determines what point in X should be

evaluated next via a proxy optimization x

next

= argmax

x

a(x),where several different functions

have been proposed.In general,these acquisition functions depend on the previous observations,

as well as the GP hyperparameters;we denote this dependence as a(x;fx

n

;y

n

g;).There are

several popular choices of acquisition function.Under the Gaussian process prior,these functions

depend on the model solely through its predictive mean function (x;fx

n

;y

n

g;) and predictive

variance function

2

(x;fx

n

;y

n

g;).In the proceeding,we will denote the best current value

as x

best

= argmin

x

n

f(x

n

),() will denote the cumulative distribution function of the standard

normal,and () will denote the standard normal density function.

Probability of Improvement One intuitive strategy is to maximize the probability of improving

over the best current value [12].Under the GP this can be computed analytically as

a

PI

(x;fx

n

;y

n

g;) = ( (x)); (x) =

f(x

best

) (x;fx

n

;y

n

g;)

(x;fx

n

;y

n

g;)

:(1)

Expected Improvement Alternatively,one could choose to maximize the expected improvement

(EI) over the current best.This also has closed formunder the Gaussian process:

a

EI

(x;fx

n

;y

n

g;) = (x;fx

n

;y

n

g;) ( (x) ( (x)) +N( (x);0;1)) (2)

GP Upper Conﬁdence Bound A more recent development is the idea of exploiting lower conﬁ-

dence bounds (upper,when considering maximization) to construct acquisition functions that mini-

mize regret over the course of their optimization [3].These acquisition functions have the form

a

LCB

(x;fx

n

;y

n

g;) = (x;fx

n

;y

n

g;) (x;fx

n

;y

n

g;);(3)

with a tunable to balance exploitation against exploration.

In this work we will focus on the EI criterion,as it has been shown to be better-behaved than

probability of improvement,but unlike the method of GP upper conﬁdence bounds (GP-UCB),it

does not require its own tuning parameter.Although the EI algorithmperforms well in minimization

problems,we wish to note that the regret formalization may be more appropriate in some settings.

We performa direct comparison between our EI-based approach and GP-UCB in Section 4.1.

3 Practical Considerations for Bayesian Optimization of Hyperparameters

Although an elegant framework for optimizing expensive functions,there are several limitations

that have prevented it from becoming a widely-used technique for optimizing hyperparameters in

machine learning problems.First,it is unclear for practical problems what an appropriate choice is

for the covariance function and its associated hyperparameters.Second,as the function evaluation

itself may involve a time-consuming optimization procedure,problems may vary signiﬁcantly in

duration and this should be taken into account.Third,optimization algorithms should take advantage

of multi-core parallelism in order to map well onto modern computational environments.In this

section,we propose solutions to each of these issues.

3.1 Covariance Functions and Treatment of Covariance Hyperparameters

The power of the Gaussian process to express a rich distribution on functions rests solely on the

shoulders of the covariance function.While non-degenerate covariance functions correspond to

inﬁnite bases,they nevertheless can correspond to strong assumptions regarding likely functions.In

particular,the automatic relevance determination (ARD) squared exponential kernel

K

SE

(x;x

0

) =

0

exp

1

2

r

2

(x;x

0

)

r

2

(x;x

0

) =

D

X

d=1

(x

d

x

0

d

)

2

=

2

d

:(4)

is often a default choice for Gaussian process regression.However,sample functions with this co-

variance function are unrealistically smooth for practical optimization problems.We instead propose

3

the use of the ARD Mat´ern 5=2 kernel:

K

M52

(x;x

0

) =

0

1 +

p

5r

2

(x;x

0

) +

5

3

r

2

(x;x

0

)

exp

n

p

5r

2

(x;x

0

)

o

:(5)

This covariance function results in sample functions which are twice-differentiable,an assumption

that corresponds to those made by,e.g.,quasi-Newton methods,but without requiring the smooth-

ness of the squared exponential.

After choosing the formof the covariance,we must also manage the hyperparameters that govern its

behavior (Note that these “hyperparameters” are distinct from those being subjected to the overall

Bayesian optimization.),as well as that of the mean function.For our problems of interest,typically

we would have D+3 Gaussian process hyperparameters:D length scales

1:D

,the covariance

amplitude

0

,the observation noise ,and a constant mean m.The most commonly advocated ap-

proach is to use a point estimate of these parameters by optimizing the marginal likelihood under the

Gaussian process,p(yj fx

n

g

N

n=1

;;;m) = N(yj m1;

+I),where y = [y

1

;y

2

; ;y

N

]

T

,

and

is the covariance matrix resulting fromthe N input points under the hyperparameters .

However,for a fully-Bayesian treatment of hyperparameters (summarized here by alone),it is

desirable to marginalize over hyperparameters and compute the integrated acquisition function:

^a(x;fx

n

;y

n

g) =

Z

a(x;fx

n

;y

n

g;) p( j fx

n

;y

n

g

N

n=1

) d;(6)

where a(x) depends on and all of the observations.For probability of improvement and EI,this

expectation is the correct generalization to account for uncertainty in hyperparameters.We can

therefore blend acquisition functions arising fromsamples fromthe posterior over GP hyperparam-

eters and have a Monte Carlo estimate of the integrated expected improvement.These samples can

be acquired efﬁciently using slice sampling,as described in Murray and Adams [13].As both opti-

mization and Markov chain Monte Carlo are computationally dominated by the cubic cost of solving

an N-dimensional linear system(and our function evaluations are assumed to be much more expen-

sive anyway),the fully-Bayesian treatment is sensible and our empirical evaluations bear this out.

Figure 1 shows how the integrated expected improvement changes the acquistion function.

3.2 Modeling Costs

Ultimately,the objective of Bayesian optimization is to ﬁnd a good setting of our hyperparameters

as quickly as possible.Greedy acquisition procedures such as expected improvement try to make

the best progress possible in the next function evaluation.From a practial point of view,however,

we are not so concerned with function evaluations as with wallclock time.Different regions of

the parameter space may result in vastly different execution times,due to varying regularization,

learning rates,etc.To improve our performance in terms of wallclock time,we propose optimizing

with the expected improvement per second,which prefers to acquire points that are not only likely

to be good,but that are also likely to be evaluated quickly.This notion of cost can be naturally

generalized to other budgeted resources,such as reagents or money.

Just as we do not know the true objective function f(x),we also do not know the duration func-

tion c(x):X!R

+

.We can nevertheless employ our Gaussian process machinery to model lnc(x)

alongside f(x).In this work,we assume that these functions are independent of each other,although

their coupling may be usefully captured using GP variants of multi-task learning (e.g.,[14,15]).

Under the independence assumption,we can easily compute the predicted expected inverse duration

and use it to compute the expected improvement per second as a function of x.

3.3 Monte Carlo Acquisition for Parallelizing Bayesian Optimization

With the advent of multi-core computing,it is natural to ask how we can parallelize our Bayesian

optimization procedures.More generally than simply batch parallelism,however,we would like to

be able to decide what x should be evaluated next,even while a set of points are being evaluated.

Clearly,we cannot use the same acquisition function again,or we will repeat one of the pending

experiments.Ideally,we could perform a roll-out of our acquisition policy,to choose a point that

appropriately balanced information gain and exploitation.However,such roll-outs are generally

intractable.Instead we propose a sequential strategy that takes advantage of the tractable inference

properties of the Gaussian process to compute Monte Carlo estimates of the acquisiton function

under different possible results frompending function evaluations.

4

(a) Posterior samples under varying hyperparameters

(b) Expected improvement under varying hyperparameters

(c) Integrated expected improvement

Figure 1:Illustration of integrated expected improve-

ment.(a) Three posterior samples are shown,each

with different length scales,after the same ﬁve obser-

vations.(b) Three expected improvement acquisition

functions,with the same data and hyperparameters.

The maximum of each is shown.(c) The integrated

expected improvement,with its maximumshown.

(a) Posterior samples after three data

(b) Expected improvement under three fantasies

(c) Expected improvement across fantasies

Figure 2:Illustration of the acquisition with pend-

ing evaluations.(a) Three data have been observed

and three posterior functions are shown,with “fan-

tasies” for three pending evaluations.(b) Expected im-

provement,conditioned on the each joint fantasy of the

pending outcome.(c) Expected improvement after in-

tegrating over the fantasy outcomes.

Consider the situation in which N evaluations have completed,yielding data fx

n

;y

n

g

N

n=1

,and in

which J evaluations are pending at locations fx

j

g

J

j=1

.Ideally,we would choose a new point based

on the expected acquisition function under all possible outcomes of these pending evaluations:

^a(x;fx

n

;y

n

g;;fx

j

g) =

Z

R

J

a(x;fx

n

;y

n

g;;fx

j

;y

j

g) p(fy

j

g

J

j=1

j fx

j

g

J

j=1

;fx

n

;y

n

g

N

n=1

) dy

1

dy

J

:(7)

This is simply the expectation of a(x) under a J-dimensional Gaussian distribution,whose mean and

covariance can easily be computed.As in the covariance hyperparameter case,it is straightforward

to use samples from this distribution to compute the expected acquisition and use this to select the

next point.Figure 2 shows howthis procedure would operate with queued evaluations.We note that

a similar approach is touched upon brieﬂy by Ginsbourger and Riche [16],but they view it as too

intractable to warrant attention.We have found our Monte Carlo estimation procedure to be highly

effective in practice,however,as will be discussed in Section 4.

4 Empirical Analyses

In this section,we empirically analyse

1

the algorithms introduced in this paper and compare to ex-

isting strategies and human performance on a number of challenging machine learning problems.

We refer to our method of expected improvement while marginalizing GP hyperparameters as “GP

EI MCMC”,optimizing hyperparameters as “GP EI Opt”,EI per second as “GP EI per Second”,and

N times parallelized GP EI MCMC as “Nx GP EI MCMC”.Each results ﬁgure plots the progres-

sion of min

x

n

f(x

n

) over the number of function evaluations or time,averaged over multiple runs

of each algorithm.If not speciﬁed otherwise,x

next

= argmax

x

a(x) is computed using gradient-

based search with multiple restarts (see supplementary material for details).The code used is made

publicly available at http://www.cs.toronto.edu/

˜

jasper/software.html.

1

All experiments were conducted on identical machines using the Amazon EC2 service.

5

(a)

(b)

(c)

Figure 3:Comparisons on the Branin-Hoo function (3a) and training logistic regression on MNIST (3b).(3c)

shows GP EI MCMC and GP EI per Second from(3b),but in terms of time elapsed.

(a)

(b)

(c)

Figure 4:Different strategies of optimization on the Online LDA problem compared in terms of function

evaluations (4a),walltime (4b) and constrained to a grid or not (4c).

4.1 Branin-Hoo and Logistic Regression

We ﬁrst compare to standard approaches and the recent Tree Parzen Algorithm

2

(TPA) of Bergstra

et al.[5] on two standard problems.The Branin-Hoo function is a common benchmark for Bayesian

optimization techniques [2] that is deﬁned over x 2 R

2

where 0 x

1

15 and 5 x

2

15.We

also compare to TPA on a logistic regression classiﬁcation task on the popular MNIST data.The

algorithmrequires choosing four hyperparameters,the learning rate for stochastic gradient descent,

on a log scale from 0 to 1,the`

2

regularization parameter,between 0 and 1,the mini batch size,

from20 to 2000 and the number of learning epochs,from5 to 2000.Each algorithmwas run on the

Branin-Hoo and logistic regression problems 100 and 10 times respectively and mean and standard

error are reported.The results of these analyses are presented in Figures 3a and 3b in terms of

the number of times the function is evaluated.On Branin-Hoo,integrating over hyperparameters is

superior to using a point estimate and the GP EI signiﬁcantly outperforms TPA,ﬁnding the minimum

in less than half as many evaluations,in both cases.For logistic regression,3b and 3c show that

although EI per second is less efﬁcient in function evaluations it outperforms standard EI in time.

4.2 Online LDA

Latent Dirichlet Allocation (LDA) is a directed graphical model for documents in which words

are generated from a mixture of multinomial “topic” distributions.Variational Bayes is a popular

paradigm for learning and,recently,Hoffman et al.[17] proposed an online learning approach in

that context.Online LDA requires 2 learning parameters,

0

and ,that control the learning rate

t

= (

0

+ t)

used to update the variational parameters of LDA based on the t

th

minibatch of

document word count vectors.The size of the minibatch is also a third parameter that must be

chosen.Hoffman et al.[17] relied on an exhaustive grid search of size 6 6 8,for a total of 288

hyperparameter conﬁgurations.

We used the code made publically available by Hoffman et al.[17] to run experiments with online

LDA on a collection of Wikipedia articles.We downloaded a random set of 249 560 articles,split

into training,validation and test sets of size 200 000,24 560 and 25 000 respectively.The documents

are represented as vectors of word counts froma vocabulary of 7702 words.As reported in Hoffman

et al.[17],we used a lower bound on the per word perplexity of the validation set documents as the

performance measure.One must also specify the number of topics and the hyperparameters for

the symmetric Dirichlet prior over the topic distributions and for the symmetric Dirichlet prior

over the per document topic mixing weights.We followed Hoffman et al.[17] and used 100 topics

and = = 0:01 in our experiments in order to emulate their analysis and repeated exactly the grid

2

Using the publicly available code fromhttps://github.com/jaberg/hyperopt/wiki

6

(a)

(b)

(c)

Figure 5:Acomparison of various strategies for optimizing the hyperparameters of M3E models on the protein

motif ﬁnding task in terms of walltime (5a),function evaluations (5b) and different covariance functions(5c).

search reported in the paper

3

.Each online LDA evaluation generally took between ﬁve to ten hours

to converge,thus the grid search requires approximately 60 to 120 processor days to complete.

In Figures 4a and 4b we compare our various strategies of optimization over the same grid on this

expensive problem.That is,the algorithms were restricted to only the exact parameter settings as

evaluated by the grid search.Each optimization was then repeated 100 times (each time picking two

different random experiments to initialize the optimization with) and the mean and standard error

are reported

4

.Figure 4c also presents a 5 run average of optimization with 3 and 5 times parallelized

GP EI MCMC,but without restricting the new parameter setting to be on the pre-speciﬁed grid (see

supplementary material for details).A comparison with their “on grid” versions is illustrated.

Clearly integrating over hyperparameters is superior to using a point estimate in this case.While

GP EI MCMC is the most efﬁcient in terms of function evaluations,we see that parallelized GP EI

MCMC ﬁnds the best parameters in signiﬁcantly less time.Finally,in Figure 4c we see that the

parallelized GP EI MCMC algorithms ﬁnd a signiﬁcantly better minimum value than was found in

the grid search used by Hoffman et al.[17] while running a fraction of the number of experiments.

4.3 Motif Finding with Structured Support Vector Machines

In this example,we consider optimizing the learning parameters of Max-Margin Min-Entropy

(M3E) Models [18],which include Latent Structured Support Vector Machines [19] as a special

case.Latent structured SVMs outperform SVMs on problems where they can explicitly model

problem-dependent hidden variables.A popular example task is the binary classiﬁcation of pro-

tein DNA sequences [18,20,19].The hidden variable to be modeled is the unknown location of

particular subsequences,or motifs,that are indicators of positive sequences.

Setting the hyperparameters,such as the regularisation term,C,of structured SVMs remains a chal-

lenge and these are typically set through a time consuming grid search procedure as is done in

[18,19].Indeed,Kumar et al.[20] avoided hyperparameter selection for this task as it was too

computationally expensive.However,Miller et al.[18] demonstrate that results depend highly on

the setting of the parameters,which differ for each protein.M3E models introduce an entropy term,

parameterized by ,which enables the model to outperformlatent structured SVMs.This additional

performance,however,comes at the expense of an additional problem-dependent hyperparameter.

We emulate the experiments of Miller et al.[18] for one protein with approximately 40 000 se-

quences.We explore 25 settings of the parameter C,on a log scale from10

1

to 10

6

,14 settings of

,on a log scale from 0.1 to 5 and the model convergence tolerance, 2 f10

4

,10

3

,10

2

,10

1

g.

We ran a grid search over the 1400 possible combinations of these parameters,evaluating each over

5 random50-50 training and test splits.

In Figures 5a and 5b,we compare the randomized grid search to GP EI MCMC,GP EI per Second

and their 3x parallelized versions,all constrained to the same points on the grid.Each algorithm

was repeated 100 times and the mean and standard error are shown.We observe that the Bayesian

optimization strategies are considerably more efﬁcient than grid search which is the status quo.In

this case,GP EI MCMC is superior to GP EI per Second in terms of function evaluations but GP

EI per Second ﬁnds better parameters faster than GP EI MCMC as it learns to use a less strict

3

i.e.the only difference was the randomly sampled collection of articles in the data set and the choice of the

vocabulary.We ran each evaluation for 10 hours or until convergence.

4

The restriction of the search to the same grid was chosen for efﬁciency reasons:it allowed us to repeat

the experiments several times efﬁciently,by ﬁrst computing all function evaluations over the whole grid and

reusing these values within each repeated experiment.

7

Figure 6:Validation error on the CIFAR-10 data for different optimization strategies.

convergence tolerance early on while exploring the other parameters.Indeed,3x GP EI per second,

is the least efﬁcient in terms of function evaluations but ﬁnds better parameters faster than all the

other algorithms.Figure 5c compares the use of various covariance functions in GP EI MCMC

optimization on this problem,again repeating the optimization 100 times.It is clear that the selection

of an appropriate covariance signiﬁcantly affects performance and the estimation of length scale

parameters is critical.The assumption of the inﬁnite differentiability as imposed by the commonly

used squared exponential is too restrictive for this problem.

4.4 Convolutional Networks on CIFAR-10

Neural networks and deep learning methods notoriously require careful tuning of numerous hyper-

parameters.Multi-layer convolutional neural networks are an example of such a model for which a

thorough exploration of architechtures and hyperparameters is beneﬁcial,as demonstrated in Saxe

et al.[21],but often computationally prohibitive.While Saxe et al.[21] demonstrate a methodology

for efﬁciently exploring model architechtures,numerous hyperparameters,such as regularisation

parameters,remain.In this empirical analysis,we tune nine hyperparameters of a three-layer con-

volutional network [22] on the CIFAR-10 benchmark dataset using the code provided

5

.This model

has been carefully tuned by a human expert [22] to achieve a highly competitive result of 18%test

error on the unaugmented data,which matches the published state of the art result [23] on CIFAR-

10.The parameters we explore include the number of epochs to run the model,the learning rate,

four weight costs (one for each layer and the softmax output weights),and the width,scale and

power of the response normalization on the pooling layers of the network.

We optimize over the nine parameters for each strategy on a withheld validation set and report the

mean validation error and standard error over ﬁve separate randomly initialized runs.Results are

presented in Figure 6 and contrasted with the average results achieved using the best parameters

found by the expert.The best hyperparameters found by the GP EI MCMC approach achieve an

error on the test set of 14:98%,which is over 3% better than the expert and the state of the art on

CIFAR-10.The same procedure was repeated on the CIFAR-10 data augmented with horizontal

reﬂections and translations,similarly improving on the expert from 11% to 9.5% test error and

achieving to our knowledge the lowest error reported on the competitive CIFAR-10 benchmark.

5 Conclusion

We presented methods for performing Bayesian optimization for hyperparameter selection of gen-

eral machine learning algorithms.We introduced a fully Bayesian treatment for EI,and algorithms

for dealing with variable time regimes and running experiments in parallel.The effectiveness of our

approaches were demonstrated on three challenging recently published problems spanning different

areas of machine learning.The resulting Bayesian optimization ﬁnds better hyperparameters sig-

niﬁcantly faster than the approaches used by the authors and surpasses a human expert at selecting

hyperparameters on the competitive CIFAR-10 dataset,beating the state of the art by over 3%.

Acknowledgements

The authors thank Alex Krizhevsky,Hoffman et al.[17] and Miller et al.[18] for making their code

and data available,and George Dahl for valuable feedback.This work was funded by DARPAYoung

Faculty Award N66001-12-1-4219,NSERC and an Amazon AWS in Research grant.

5

Available at:http://code.google.com/p/cuda-convnet/

8

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